Alternative Design for Quantum Cryptographic Entangling Probe

ABSTRACT

An alternative design is given for an optimized quantum cryptographic entangling probe for attacking the BB84 protocol of quantum key distribution. The initial state of the probe has a simpler analytical dependence on the set error rate to be induced by the probe than in the earlier design. The new device yields maximum information to the probe for a full range of induced error rates. As in the earlier design, the probe contains a single CNOT gate which produces the optimum entanglement between the BB84 signal states and the correlated probe states.

RELATED APPLICATIONS

This application is related to and claims benefit of U.S. patent application Ser. No. 11/239,461 filed 16 Sep. 2005 which, in urn claims benefit of Provisional Application 60/617,796 filed 9 Oct. 2004. This invention is a continuation-in-part and an improvement on the earlier application by the same inventor that is incorporated herein in its entirety by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to cryptographic entangling probes used in communications.

2. Description of the Prior Art

Recently, a design was given by H. E. Brandt, “Quantum cryptographic entangling probe,” Phys. Rev. A 71, 042312(14) (2005) [1]. H. E. Brandt, “Design for a quantum cryptographic entangling probe,” J. Modern Optics 52, 2177-2185 (2005) [2] for an optimized entangling probe attacking the BB84 Protocol, C. H. Bennett and G. Brassard, Quantum cryptography: “public key distribution and coin tossing”, Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, New York, 1984), pp. 175-179 [3] of quantum key distribution (QKD) and yielding maximum Renyi information to the probe for a set error rate induced by the probe. Probe photon polarization states become optimally entangled with the BB84 signal states on their way between the legitimate transmitter and receiver. Standard von Neumann projective measurements of the probe yield maximum information on the pre-privacy amplified key, once basis information is revealed during reconciliation. A simple quantum circuit was found, consisting of a single CNOT gate, and faithfully producing the optimal entanglement. The control qubit consists of two photon polarization-basis states of the signal, the target qubit consists of two probe photon polarization basis states, and the initial state of the probe is set by an explicit algebraic function of the error rate to be induced by the probe. A method was determined for measuring the appropriate probe states correlated with the BB84 signal states and yielding maximum Renyi information to the probe. The design presented in [1], [2] was limited to error rates not exceeding ¼, but is generalized in the instant invention to allow a full range of error rates from 0 to ⅓, H. E. Brandt and J. M. Myers, J. Mod. Optics, 53, 1927-1930 (2006) [4].

SUMMARY OF THE INVENTION

Accordingly, an object of this invention is an improved probe.

These and additional objects of the invention are accomplished by a new and simpler probe design for which the induced error rate also ranges from 0 to ⅓. It is based on an alternative optimum unitary transformation in [1], also yielding the same maximum Renyi information to the probe. The initial state of the probe has a simpler algebraic dependence on the set error rate to be induced by the probe. The alternative optimized unitary transformation is reviewed, representing the action of an optimized entangling probe yielding maximum information on quantum key distribution in the BB84 protocol. In the Section QUANTUM CIRCUIT AND PROBE DESIGN that follows, the quantum circuit and design are given for the new entangling probe. The concluding Section contains a summary.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the present work, an implementation was determined of the optimum unitary transformation given by Eqs. (158)-(164) of [1], however with restricted parameters such that the corresponding Hilbert space of the probe reduces from four to two dimensions. In particular, the parameters μ and θ are here restricted to

sin μ=cos μ=2^(−1/2); cos θ=1:   (1)

In this case, the entangling probe states|94

, |σ⁻

, |σ

, |δ

,|δ⁻

, |δ

, given by Eqs. (159)-(164) of [1], become

|σ

=|δ⁻

=4[(1−2E)^(1/2)|w₀

⁻ E ^(1/2)|w_(b)

],   (2)

|σ

=|δ⁻

=4[(1−2E)^(1/2)|w_(a)

+E ^(1/2)|w_(b)

],   (3)

|σ

=−|δ

==4E ^(1/2)|w_(b)

,   (4)

in which the upper sign choices in Eqs. (159)-(164) of [1] have been made, E is the error rate induced by the probe, and the orthonormal probe basis vectors |w_(a)

and |w_(b)

are defined by

|w_(a)

=2^(−1/2)(|w₀

+|w₃

,   (5)

|w_(b)

=2^(−1/2)(|w₁

−|w₂

),   (6)

expressed in terms of the orthonormal basis vectors |w_(a)

,|w₃

, |w

, and |w₂

of [1]. the optimum unitary transformation, Eq. (158) of [1] produces in this case the following entanglements for initial probe state |w

and incoming BB84 signal photon-polarization states |u

, |ū

, |v

, or | v

, respectively:

$\begin{matrix} {{{{u\rangle} \otimes {w\rangle}}->{\frac{1}{4}\left( {{{u\rangle} \otimes {\sigma_{+}\rangle}} + {{\overset{\_}{u}\rangle} \otimes {\sigma\rangle}}} \right)}},} & (7) \\ {{{{\overset{\_}{u}\rangle} \otimes {w\rangle}}->{\frac{1}{4}\left( {{{u\rangle} \otimes {\sigma\rangle}} + {{\overset{\_}{u}\rangle} \otimes {\sigma_{-}\rangle}}} \right)}},} & (8) \\ {{{{v\rangle} \otimes {w\rangle}}->{\frac{1}{4}\left( {{{v\rangle} \otimes {\sigma_{-}\rangle}} - {{\overset{\_}{v}\rangle} \otimes {\sigma\rangle}}} \right)}},} & (9) \\ {{{{\overset{\_}{v}\rangle} \otimes {w\rangle}}->{\frac{1}{4}\left( {{{- {v\rangle}} \otimes {\sigma\rangle}} + {{\overset{\_}{v}\rangle} \otimes {\sigma_{-}\rangle}}} \right)}},} & (10) \end{matrix}$

Here, the probe states |σ₊

, |σ⁻

, |σ

are given by Eqs. (2)-(4). The states |u

and |ū

are orthogonal linearly-polarized photon signal states in the {|u

, |ū

} basis, and |v

and | v

are orthogonal linearly-polarized photon signal states in the {|v

, | v

} basis, and the two bases are nonothogonal with π/4 angle between the linear polarizations of states |u

and |v

. In the present case, the maximum information gain by the probe is again given by

$\begin{matrix} {{I_{opt}^{R} = {\log_{2}\left\lbrack {2 - \left( \frac{1 - {3E}}{1 - E} \right)^{2}} \right\rbrack}},} & (11) \end{matrix}$

and here E≦⅓, since E=⅓, according to Eq. (11), corresponds to perfect information gain by the probe.

Quantum Circuit and Probe Design

Using the same methods presented in [1], it can be shown that a quantum circuit consisting again of a single CNOT gate suffices to produce the optimum entanglement, Eqs. (7)-(10). Here, the control qubit entering the control port of the CNOT gate consists of the two signal basis states {|e₀

, |e₁

}. In the two-dimensional Hilbert space of the signal, the basis states |e₀

and |e₁

, respectively, are orthonormal and make equal angles of π/8 with the nonorthogonal signal states |u

and |v

, respectively. The target qubit entering the target port of the CNOT gate consists of the two orthonormal linearly-polarized photon polarization computational basis states 2^(−1/2) (|w_(a)

±|w_(b)

). When |e₀

enters the control port, {|w_(a)

, |w_(b)

} becomes {|w_(a)

, −|w_(b)

}, and when |e₁

enters the control port, {|w_(a)

, |w_(b)

} remains the same. The initial unnormalized target state of the probe can, in this case, be shown to be given by:

|A ₂

=(1−2E)^(1/2)|w _(a)

+(2E)^(1/2)|w _(b)

,   (12)

and the unnormalized transition state is given by

|A ₁

=(1−2E)^(1/2)|w _(a)

−(2E)^(1/2)|w _(b)

.   (13)

Next, by arguments directly paralleling those of [1], using Eqs. (7)-(10), one has the following correlations between the signal states and the projected probe states, |σ₃₀

and |σ⁻

:

|u

|σ₊

|ū

σ⁻

  (14)

and

|v

σ−

| v

  (15)

The measurement basis for the symmetric von Neumann projective measurement of the probe must be orthogonal and symmetric about the correlated probe states, |σ₊

and |σ−

in the two-dimensional Hilbert space of the probe [1]. Thus, consistent with Eqs. (2) and (3), I define the following orthonormal measurement basis states:

|w ₊

₌2^(−1/2)(|w _(a)

+|w _(b)

),   (16)

|w ⁻

=2^(−1/2)(|w _(a)

−|w _(b)

).   (17)

Next, one notes that the correlations of the projected probe states |σ₊

and |σ⁻

with the measurement basis states |w₊

and |w⁻

are indicated, according to Eqs. (2), (3), (16), and (17), by the following probabilities:

$\begin{matrix} {{\frac{{{\langle\left. w_{+} \middle| \sigma_{+} \right.\rangle}}^{2}}{{\sigma_{+}}^{2}} = {\frac{{{\langle\left. w_{-} \middle| \sigma_{-} \right.\rangle}}^{2}}{{\sigma_{-}}^{2}} = {\frac{1}{2} - \frac{{E^{1/2}\left( {1 - {2E}} \right)}^{1/2}}{\left( {1 - E} \right)}}}},} & (18) \\ {{\frac{{{\langle\left. w_{+} \middle| \sigma_{-} \right.\rangle}}^{2}}{{\sigma_{-}}^{2}} = {\frac{{{\langle\left. w_{-} \middle| \sigma_{+} \right.\rangle}}^{2}}{{\sigma_{+}}^{2}} = {\frac{1}{2} + \frac{{E^{1/2}\left( {1 - {2E}} \right)}^{1/2}}{\left( {1 - E} \right)}}}},} & (19) \end{matrix}$

consistent with Eqs. (198) and (199) of [1], and implying the following dominant state correlations:

|σ₊

w⁻

, |σ⁻

w₊

.   (20)

Next combining the correlations (14), (15), and (20), one thus establishes the following correlations:

{|u

,| v

}

σ₊

w⁻

.   (21)

{|ū

, |v

}

σ⁻

|w₊

.   (22)

to be implemented by the projective measurement of the probe, as in [1]. One therefore arrives at the following alternative entangling probe design. An incident photon coming from the legitimate transmitter is received by the probe in one of the four signal-photon linear-polarization states |u

, |ū

, |v

, or | v

in the BB84 protocol. The signal photon enters the control port of a CNOT gate. The initial state of the probe is a photon in linear-polarization state |A₂

entering the target port of the CNOT gate. The probe photon is produced by a single-photon source and is appropriately timed with reception of the signal photon by first sampling a few successive signal pulses to determine the repetition rate of the transmitter. The photon linear-polarization state |A₂

, according to Eq. (12), is given by

|A₂

=(1−2E)^(1/2) |w _(a)

+(2E)^(1/2) |w _(b)

,   (23)

and can be simply set for an error rate E by means of a polarizer. (Note that this initial probe state has a simpler algebraic dependence on error rate than that in [1] or [4]).

In accord with Eq. (23), the entangling probe can be tuned to the chosen error rate to be induced by the probe. The outgoing gated signal photon is relayed on to the legitimate receiver, and the gated probe photon enters a Wollaston prism, oriented to separate photon orthogonal-linear-polarization states |w₊

and |w⁻

, and the photon is then detected by one of two photodetectors. This is an ordinary symmetric von Neumann projective measurement. If the basis, revealed during the public basis-reconciliation phase of the BB84 protocol, is {|u

, |ū

}, then the photodetector located to receive the polarization state |w⁻

or |w₊

, respectively, will indicate, in accord with the correlations (21) and (22), that a state |u

or |ū

, respectively, was most likely measured by the legitimate receiver.

Alternatively, if the announced basis is {|v

, | v

}, then the photodetector located to receive the polarization state |w₃₀

or |w⁻

, respectively, will indicate, in accord with the correlations (21) and (22), that a state |v

or | v

, respectively, was most likely measured by the legitimate receiver. By comparing the record of probe photodetector triggering with the sequence of bases revealed during reconciliation. then the likely sequence of ones and zeroes constituting the key, prior to privacy amplification, can be assigned. In any case the net effect is to yield, for a set error rate E, the maximum Renyi information gain to the probe, which is given by Eq. (11).

CONCLUSION

An alternative design is given for an optimized quantum cryptographic entangling probe for attacking the BB84 protocol of quantum key distribution. The initial state of the probe has a simpler analytical dependence on the set error rate to be induced by the probe. The device yields maximum information to the probe for a full range of induced error rates. As in the earlier design, the probe contains a single CNOT gate which produces the optimum entanglement between the BB84 signal states and the correlated probe states, and is measured by making a symmetric von Neumann projective test.

Obviously, many modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that, within the scope of the appended claims, the invention many be practiced otherwise than as specifically described. 

1. A system for obtaining information on a key in the BB84 protocol of quantum key distribution, comprising: a quantum cryptographic entangling probe comprising; a single-photon source configured to produce a probe photon; a polarization filter configured to determine an optimum initial probe photon polarization state for a set error rate induced by the quantum cryptographic entangling probe; a quantum controlled-NOT (CNOT) gate configured to provide entanglement of a signal with the probe photon polarization state and produce a gated probe photon so as to obtain information on a key; a Wollaston polarization beam splitter prism configured to separate the gated probe photon with polarization correlated to a signal measured by the legitimate receiver; and two single-photon photodetectors configured to measure the polarization state of the gated probe photon.
 2. The system of claim 1, wherein the quantum CNOT gate is further configured to provide optimum entanglement of the signal with the probe photon polarization state to produce the gated probe photon so as to obtain maximum Renyi information from the signal.
 3. The system of claim 1, wherein the CNOT gate is a cavity QED implementation.
 4. The system of claim 1, wherein the CNOT gate is a quantum dot implementation.
 5. The system of claim 1 wherein the CNOT gate is a solid state implementation.
 6. The system of claim l, wherein the CNOT gate is a linear optics implementation.
 7. A method for obtaining information on a key in the BB84 protocol of quantum key distribution, said method comprising: configuring a single-photon source for producing a probe photon; determining an initial probe photon polarization state corresponding to a set error rate induced by a probe; entangling a signal with a probe photon polarization state and producing a gated probe photon so as to obtain information on a key; separating the gated probe photon with polarization correlated with a signal measured by a receiver; measuring the polarization state of the gated probe photon; accessing information on polarization-basis selection available on a public classical communication channel between the transmitter and the receiver; and determining the polarization state measured by the receiver.
 8. The method of claim 7, wherein the entangling comprises entangling the signal with the probe photon polarization state so as to obtain maximum Renyi information from the signal,
 9. The method of claim 7, further comprising receiving a signal photon from a transmitter prior to the configuring of the single-photon source.
 10. The method of claim 7, further comprising relaying an outgoing gated signal photon to the receiver.
 11. The method of claim 7, wherein the separating utilizes a Wollaston prism.
 12. The method of claim 7, wherein the separating utilizes a polarization beam splitter.
 13. The method of claim 7, wherein the single-photon source is synchronized with the signal photons from. the transmitter by first sampling successive signal pulses to determine a repetition rate of the transmitter.
 14. A system for obtaining information on a key in the BB84 protocol of quantum key distribution, comprising: means for configuring a single-photon source to produce a probe photon; means for determining an optimum initial probe photon polarization state corresponding to a set error rate induced by a probe; means for entangling a signal with a probe photon polarization state and producing a gated probe photon so as to obtain information on a key; means for separating the gated probe photon with polarization correlated with a signal measured by a receiver; means for measuring the polarization state of the gated probe photon; means for accessing information on polarization-basis selection available on a public classical communication channel between the transmitter and the receiver; and means for determining the polarization state measured by the receiver.
 15. The system of claim 14, further comprising means for receiving a signal photon from a transmitter.
 16. The system of claim 14, further comprising means for relaying an outgoing signal photon to a receiver.
 17. The system of claim 14, further comprising means for providing entanglement of the signal with the probe photon polarization state so as to obtain maximum Renyi information from the signal.
 18. The system of claim 14, further comprising means for timing the single photon source by receiving a signal photon from a transmitter by first sampling successive signal pulses to determine a repetition rate of the transmitter. 